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ESTIMATING CLEAR-SKY BEAM IRRADIATION FROM SUNSHINEDURATION

HELEN C. POWER†University of South Carolina, Department of Geography, Columbia, SC 29208, USA

Received 15 August 2000; revised version accepted 18 April 2001

Communicated by AMOS ZEMEL

Abstract—Monthly-averaged climate and turbidity data from five sites in North America and Europe are¯ ¯employed to evaluate the relationships among monthly-averaged observed beam irradiation (H and H ),b bn

¯ ¯ ¯¯clear-sky beam irradiation (H and H ), sunshine duration (s ), and daylength (S ). Linear regression modelbc bnc¯ ¯ ¯ ¯ ¯ ¯¯ ¯fits to H /H 5 a 1 b(s /S ) and H /H 5 a 1 b(s /S ) indicate that the intercept (a) and slope (b) of the fitsb bc bn bnc

are close to 0.0 and 1.0, respectively. Thus, in locations where the absence of empirical data prevents findingthe ‘true’ values of these coefficients, it is proposed that a50 and b51 be used to estimate the clear-sky beamirradiation from observed irradiation and sunshine duration. The analysis has utility in turbidity studies andsolar applications, such as the performance prediction of solar energy systems, where the average dailyclear-sky beam irradiation needs to be known. 2001 Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION days that are completely cloud free. Thus, it is notroutinely measured.

Solar radiation information is needed in a varietyThis paper explores methods for quantifying

of applications including agriculture, water re-the potential clear-sky beam irradiation from

sources, daylighting and architectural design,observed irradiation and bright sunshine duration.

solar thermal devices, and climate change studies.To this end, climatic and turbidity data are drawn

Compared to meteorological parameters such asfrom five locations in North America and Europe.

precipitation, temperature, and wind, observedThese analyses were specifically motivated by the

irradiance data are scarce. For this reason, manyneed for quantifying the potential irradiation

different methods have been used to estimatewhich, in turn, could be used in turbidity studies.

irradiation from sunshine duration records, sinceHowever, the analyses presented here may also be

the latter are more readily available.of utility in other solar applications where the

Specific solar radiation information is alsopotential flux density needs to be known; for

needed as input to models that estimate atmos-instance, in cooling load calculations, in passive

pheric turbidity, which is a measure of the totalsolar design to avoid possible overheating, and in

amount of aerosol integrated through the verticalthe performance prediction of solar energy sys-‡atmospheric column. Gueymard (1993) andtems (see, for example, Suehrcke and McCormick

Power (2001), for example, both offer parame-(1992) and Suehrcke (2000)).

terized irradiation models that can be inverted toestimate atmospheric turbidity, but these modelsassume that the daily clear-sky beam irradiation

2. BACKGROUND(and other climate variables) are known. As

By definition, bright sunshine duration (s) is thedistinct from the observed beam irradiation, thenumber of hours per day that the sunshine intensi-daily clear-sky beam irradiation is the beamty exceeds some predetermined threshold ofradiation that would have been incident if the sky

˚ ¨brightness. Angstrom (1924) first proposed ahad been clear all day. It is, in other words, alinear relationship between the ratio of monthly-‘potential’ flux density that is only observed on

¯averaged global irradiation (H ) to cloudless glob-al irradiation (H ) and monthly-averaged sun-cg

¯shine duration (s ). It is†Tel.: 11-803-777-5867; fax: 11-803-777-4972;

e-mail: power@sc.edu ¯ ¯H s‡Some investigators define turbidity as the radiative effect of ] ]5 c 1 (1 2 c ) (1)1 1 ¯H Scgboth aerosols and water vapor.

217

218 H. C. Power

¯ ¯where c 5 0.25 and S is the monthly-averaged where H is the monthly-averaged daily beam1 b¯˚ ¨astronomical day duration (daylength). Angstrom irradiation on a horizontal surface, and H is thebc

determined the value of c from Stockholm data, monthly-averaged potential daily clear-sky beam1

but it was not until more than 30 years later irradiation on a horizontal surface. This same˚ ¨(Angstrom, 1956) that he stated that Eq. (1) was relationship was subsequently used to predict the

obtained from mean monthly data and should not performance of a solar hot water systembe used with daily data. (Suehrcke and McCormick, 1992). Hinrichsen

¯To estimate H from sunshine records, (1994) employed Eq. (3) to assign physical˚ ¨Angstrom’s model required measurements of meaning to coefficients c and c in Eq. (2), while2 3

global radiation on completely clear days (H ). Suehrcke (2000) used Eq. (3) to derive his non-cg

This limitation prompted Prescott (1940) to de- linear relationship between global radiation andvelop a model that was a function of the extrater- sunshine duration.

¯restrial radiation on a horizontal surface (H ), The physical arguments these authors made for0¯rather than H , since H can be easily calculated. horizontal irradiation suggest that the same rela-cg 0

˚ tionship exists for irradiation at normal incidence.¨This modified Angstrom equation, referred to as˚ Indeed, Gueymard (1993) proposed that¨the Angstrom–Prescott formula (Martinez-Lozano

et al., 1984; Gueymard et al., 1995), isH̄ s̄bn]] ]5 (4)¯ ¯H s ¯ ¯H Sbnc c] ]5 c 1 c (2)S D2 3¯ ¯H S0

¯where H is the monthly-averaged daily beambnwhere the overbars denote monthly average val- ¯irradiation at normal incidence, H is the month-bncues, and where c 5 0.22 and c 5 0.54, deter-2 3 ly-averaged potential daily clear-sky beam irradia-mined empirically by Prescott (1940). ¯tion at normal incidence, and S is the monthly-c

Since Prescott (1940), many empirical models averaged daylength modified to account for when†have been developed that estimate global, beam, the sun is above a critical solar elevation angle.

and diffuse radiation from the number of bright The basis of Eqs. (3) and (4) is that, for a given¨sunshine hours (e.g. Lof et al., 1966; Rietveld, day, the beam radiation incident at the surface (Hb

1978; Hay, 1979; Iqbal, 1979; Benson et al., or H ) is a fraction (s /S) of what would havebn1984; Ahmad et al., 1991; Hamdan and Al-Sayeh, been incident if the sky had been clear all day. In1991). However, these models typically utilize the absence of clouds, H and H are a functionbc bnccoefficients that are site-specific and/or dependent of atmospheric scattering and absorption pro-on the averaging period used. This limits their cesses. The appeal of Eqs. (3) and (4) is twofold:application to stations where the values of the they provide a means of estimating the potentialcoefficients were actually determined, or, at best, beam irradiation, and they do not contain em-to localities of similar climate, and for the same pirically-derived coefficients. However, a mini-averaging period. Hay (1979) lessened the spatial mum averaging period is recommended whenand temporal dependence of the coefficients by using these equations to estimate potential beamincorporating the effects of multiple reflection but irradiation; Gueymard (1993) suggested monthly.his technique requires surface and cloud albedo Time averaging is necessary since s is simplydata. More recently, Suehrcke (2000) has argued the total number of sunshine hours per day andthat the relationship between global radiation and provides no information about when the sky wassunshine duration is approximately quadratic and cloudless during any given day. Consider a hypo-thus the linear Eqs. (1) and (2) are of the wrong thetical day, for example, where S was 12 h and sfunctional form. was 6 h. If the 6 cloudless hours fell around solar

In the present study, the interest is in quantify- noon, H /H (or H /H ) will be much greaterb bc bn bncing potential beam irradiation, rather than global than if there were 3 cloudless hours in theor diffuse. However, few authors have considered morning and 3 cloudless hours in the late after-the relationships between sunshine duration, ob- noon. Eqs. (3) and (4) also make several otherserved irradiation, and the potential daily clear- assumptions: turbidity and precipitable water aresky beam irradiation. Suehrcke and McCormick the same during cloudless and partly cloudy days,(1989) first proposed the relationship measurements of s are accurate, and the sunshine

H̄ s̄b † ¯ ¯ ¯¯ ¯] ] The ratio s /S is similar to s /S in Eqs. (1)–(3) except that S5 (3) c c¯ ¯H S corrects for the irradiance threshold of sunshine recorders.bc

Estimating clear-sky beam irradiation from sunshine duration 219

recorder threshold irradiance is constant and efficacy of using observed irradiation and sun-known (Gueymard, 1993). shine duration to estimate potential irradiation on

Gueymard (1993) tested Eq. (4) using monthly both a horizontal surface and at normal incidenceaverages of H and s observed at Albany, New is examined below.bn

York, together with a parameterized model toestimate monthly means of H . For a ‘synthetic’bnc 3. ANALYSISyear of 12 months (aggregated from 26 months of

¯ ¯data), observed H compared well with H Monthly averages of irradiation ratios (H /Hbn bn b bc¯ ¯ ¯estimated from Eq. (4); the mean bias error was ¯and H /H ) and sunshine ratios (s /S ) werebn bnc

23.6% of the mean observed monthly-averaged computed, and compared, for the cities of Uccle,H , and the root mean squared error (RMSE) Belgium; Albany, New York; Montreal, Van-bn

was 5.9%. couver, and Toronto. These sites represent a rangeLouche et al. (1991) also considered the rela- of climatic environments, including marine west

tionship between H /H and s from observa- coast and humid continental. In Montreal andbn bnc

tions at Ajaccio, France. Fitting 5 years of month- Toronto, global and diffuse irradiances werely-averaged irradiation and sunshine data pro- measured with Kipp and Zonen CM5 pyranome-vided the empirical equation ters; at the remaining stations, Eppley Normal

Incidence Pyrheliometers were used. Bright sun-H̄ s̄bn shine duration (s) was measured with Campbell–]] ]5 c 1 c (5)S D4 5¯ ¯H Sbnc Stokes sunshine recorders. These instrumentshave an expected RMSE of |4.1% of the monthlywhere c 5 0.006 and c 5 0.991. Monthly means4 5percent possible sunshine (Benson et al., 1984).of H were computed using the parameterizedbncAll stations are either first class research sites ormodel of Louche et al. (1987). For this linearpart of a well-maintained monitoring network.regression, the correlation coefficient was 0.972

For Albany, Uccle, and Vancouver (where¯ ¯while the RMSE associated with H /H wasbn bnc ¯observed H were available) the mean monthly¯ ¯ bn0.036 for an average H /H of |0.55 (databn bnc ¯potential clear-sky beam normal irradiation (H )bncwere presented graphically, and without summarywas estimated using the parameterized model ofstatistics).Power (2001), namely,Both Gueymard (1993) and Power (2001) offer

a4¯parameterized irradiation models that can be paˆ 3¯ ¯ ¯ ¯ ]H 5 kE a exp(a b ) exp(a w )u S Dbnc 0n 0 1 2 oinverted to estimate atmospheric turbidity from p0H or H . It is arguable that the former type of 7bc bnc

a 225model has greater utility since, in many places, ¯3 u O s J (J m ) (7)n i ii50there are no pyrheliometric data available and so

daily beam irradiation must often be computed as where k 5 86,400/p (s), E is the extraterrestrial0nthe difference between the observed daily global irradiance at normal incidence and actual earth–

22and daily diffuse irradiation. By definition, this ˚ ¨sun distance (W m ), b is Angstrom’s turbidity†difference provides an estimate of H as opposedb coefficient, w is precipitable water (cm), u iso

to H . Thus, inverting the H models to estimatebn bc total column ozone (atm-cm), p is atmosphericturbidity dictates that H (rather than H ) bebc bnc pressure (mb), p is the sea-level standard pres-0determined from observed daily beam irradiation sure (mb), u is total column nitrogen dioxidenon a horizontal surface (H ) and sunshine dura-b (atm-cm), a and s are coefficients (Table 1), andi ition. Although Suehrcke and McCormick (1989) the overbars denote monthly averages. The valueproposed Eq. (3), it is possible that the relation- of J in Eq. (7) isi¯¯ship between H /H and s /S is of a formb bc

v0equivalent to Eq. (5), i.e. thatiJ 5E sin h dv (8)iH̄ s̄b

0] ]5 a 1 b (6)S D¯ ¯H Sbc where h is the solar elevation angle, v is the hourwhere a ± 0 and b ± 1.0.

It appears that few authors have evaluated¯ ¯experimentally the relationships between H /H † ˚b bc ¨Angstrom’s turbidity coefficient (b ) is a dimensionless¯ ¯ ¯ ¯¯ ¯and s /S, and between H /H and s /S. Given measure of turbidity defined as the aerosol optical depth atbn bnc

¯ ¯ ˚ ¨a wavelength of 1.0 mm (Angstrom, 1929).the need to quantify both H and H , thebc bnc

220 H. C. Power

Table 1. Coefficients a and s for use in Eqs. (7) and (9). Alli i incidence and actual earth-sun distance (E ), and0ncoefficients are dimensionless except a , which has units of221 the hour angle at sunrise (v ) were calculated for0cm

†the climatologically average day of each month.i a si i ¯Having estimated H in Montreal and Toronto,bc0 0.854402 0.023350 ¯and H in Albany, Uccle, and Vancouver, the1 21.913974 3.488440 bnc¯ ¯2 20.029282 26.135995 corresponding monthly irradiation ratios (H /Hb bc

3 20.012644 4.858565 ¯ ¯ ¯¯and H /H ) and sunshine ratios (s /S ) werebn bnc4 20.119903 20.709746computed from observed mean monthly irradia-5 20.002453 21.328228

6 1.007379 tion and sunshine data (Gueymard, 1999). The7 20.236401 ¯monthly-averaged daylength (S ) was calculated

for the climatologically average day of eachmonth. Linear regression models of the form

H̄ s̄bangle, and v is the hour angle at sunrise. ] ]0 5 a 1 b (10)S D¯ ¯H SAnalytic expressions for J in Eq. (8) (in terms of bci

v ) were initially proposed by Gueymard (1993)0 for Montreal and Toronto, andand are also provided in Power (2001).

¯ ¯For Montreal and Toronto (where observed H H s̄b bn]] ]5 a 1 b (11)S Dwere available), the mean monthly potential clear- ¯ ¯H Sbnc¯sky beam irradiation on a horizontal surface (H )bcfor Albany, Uccle, and Vancouver were then fit towas estimated from Power’s (2001) model:each set of n512 monthly data at each site, as

a4p̄aˆ well as to data combined from all sites (n560).3¯ ¯ ¯ ¯ ]H 5 kE a exp(a b ) exp(a w )u S Dbc 0n 0 1 2 o p0

7a 22 4. RESULTS AND DISCUSSION5¯3 u O s J (J m ). (9)n i i11

i50Across the five stations, the sunshine duration

¯¯ratios (s /S ) range between 0.196 and 0.641 (Figs.The distinction between the H model (Eq. (7))bnc

1–6). This range in values is a reflection of theand the H model (Eq. (9)) is a change in thebc

different cloud regimes that exist at each station;subscript on the J function from i to i 1 1, and¯¯the closer s /S is to zero, the more cloudy thethus an equivalent change in the exponent of the

location whereas a higher ratio is indicative of asine function in Eq. (8).less cloudy environment. Based on the globallyAs input to Eqs. (7)–(9), long-term monthly

¯¯averaged s /S, Uccle appears to be the cloudiestmeans of b and w for each station were obtained¯¯location (mean s /S 5 0.354), while Albany ap-from Gueymard (1999). Compared to b and w,

¯¯pears to be the least cloudy (mean s /S50.466).the effect of u , p, and u on surface insolation iso n

More importantly, there is clearly a linearsecondary (Iqbal, 1983). Thus, estimates (ratherrelationship between the irradiation ratios andthan direct observations) of these variables weresunshine ratios for these five locations (Table 2;deemed adequate for the present analyses. Ac-Figs. 1–6). The intercepts of the linear fits (a) arecordingly, mean monthly u at each site waso

all close to zero, ranging between 20.037 andestimated using the empirical algorithm proposed0.142, while the slopes of the fits (b) are close toby Van Heuklon (1979) as a function of latitude,1.0 and range between 0.743 and 1.002. Thelongitude, and the middle day of the month.

2coefficients of determination (R ) range betweenAtmospheric pressure ( p) at each location was0.812 and 0.994. For all sites combined (n 5 60),estimated as a function of station altitude using

2a 5 0.041, b 5 0.923, and R 5 0.865 (Table 2,the algorithm of Gueymard (1993). AppropriateFig. 6). Goodness of fit also appears to bevalues of u were estimated according to then

23 independent of whether normal incidence irradia-population of each location. A value of 2.0310atm-cm was assigned to those cities with apopulation over 1 million (Montreal, Toronto, and

23Vancouver; United Nations, 1999); 1.5310†atm-cm was assigned to Uccle, while a value of The climatologically average day of the month is based on the

241.5310 atm-cm was assigned to the smaller average solar declination and earth–sun distance for themonth.locale of Albany (see Schroeder and Davies,

1987). The extraterrestrial irradiance at normal

Estimating clear-sky beam irradiation from sunshine duration 221

¯Fig. 4. Scatterplot of monthly-averaged irradiation ratio (H /bn¯Fig. 1. Scatterplot of monthly-averaged irradiation ratio (H /bn ¯ ¯¯H ) with monthly-averaged sunshine duration ratio (s /S ) forbnc¯ ¯¯H ) with monthly-averaged sunshine duration ratio (s /S ) forbnc Uccle, Belgium; n 5 12.Albany, New York; n 5 12.

¯ ¯Fig. 2. Scatterplot of monthly-averaged irradiation ratio (H / Fig. 5. Scatterplot of monthly-averaged irradiation ratio (H /b bn¯ ¯ ¯ ¯¯ ¯H ) with monthly-averaged sunshine duration ratio (s /S ) for H ) with monthly-averaged sunshine duration ratio (s /S ) forbc bnc

Montreal; n 5 12. Vancouver; n 5 12.

¯Fig. 6. Scatterplot of monthly-averaged irradiation ratio (H /b¯ ¯ ¯¯ H or H /H ) with monthly-averaged sunshine durationFig. 3. Scatterplot of monthly-averaged irradiation ratio (H / bc bn bncb

¯¯ ¯ ¯¯ ratio (s /S ) for Uccle, Belgium; Albany, New York; Montreal,H ) with monthly-averaged sunshine duration ratio (s /S ) forbc

Vancouver, and Toronto; n 5 60.Toronto; n 5 12.

222 H. C. Power

2Table 2. Number of data points (n), intercept (a), slope (b), coefficient of determination (R ), index of agreement (d), andsystematic mean squared error (MSE ) as a percentage of total mean squared error (MSE) for linear fits of irradiation ratios as as

¯ ¯¯ ¯function of s /S where s is the monthly-averaged sunshine duration and S is the monthly-averaged astronomical day duration(daylength)

2Location n a b R d MSE /MSE Irradiations

(%) ratio

Albany 12 0.076 0.867 0.879 0.817 32.56 H /Hbn bnc

Montreal 12 0.142 0.743 0.812 0.720 56.76 H /Hb bc

Toronto 12 0.085 0.845 0.901 0.833 34.22 H /Hb bc

Uccle 12 20.037 1.002 0.994 0.758 96.81 H /Hbn bnc

Vancouver 12 0.090 0.828 0.890 0.800 39.67 H /Hbn bnc

All sites 60 0.041 0.923 0.865 0.806 8.52 Both

¯ ¯tion ratios (H /H ) or horizontal irradiation where the data sit evenly below the perfect-pre-bn bnc¯ ¯ ¯¯ratios (H /H ) are regressed against s /S. diction line. This systematic error was not evidentb bc

2At Albany, Montreal, Vancouver, and Toronto, in the high R value for this site (0.994) since thethere is a fairly even scatter of points on or about coefficient of determination is a measure ofthe perfect-prediction line (Figs. 1–3 and 5); in precision, rather than accuracy (Willmott, 1984).other words, the difference between the irradiation Overall, it appears there is a linear relationship

¯ ¯ ¯ ¯ ¯ ¯¯ ¯ratios and s /S appears fairly random. Uccle, in between H /H and s /S, and between H /Hb bc bn bnc¯ ¯ ¯¯contrast, has a different response in that H /H and s /S at these five stations. Furthermore, thebn bnc

¯¯is systematically less than s /S although the differ- data in this analysis fall close to a 1:1 relation-ences are small (Fig. 4). Interestingly, Uccle has ship. Based on the physical relationships between

¯¯the smallest range in s /S and the lowest mean irradiation and sunshine duration, it is arguable¯ ¯ ¯ ¯¯ ¯s /S. The relationship between H /H and s /S that, at least on a monthly basis, the ‘theoretical’bn bnc

is still strongly linear; a 5 2 0.037, b 5 1.002, values of the coefficients should be a50.0 and2and R 5 0.994. b51.0, as proposed by Suehrcke and McCormick

2Although R provides a measure of how well (1989) and Gueymard (1993). With regard to thethe linear models (Eqs. (10) and (11)) fit the data, empirical data in Table 2 and Figs. 1–6, thethis coefficient does not illustrate how much of deviation from the 1:1 relationship may be due to

¯ ¯ ¯ ¯the error in H /H (or H /H ), as estimated experimental error, sampling biases, uncertaintyb bc bn bnc¯¯from s /S, is systematic or unsystematic. Further- in the input data (e.g. the estimated pressure, total

2more, the R statistic is insensitive to additive and column ozone, and nitrogen dioxide), or possibleproportional differences that may exist between discrepancies in the time series of the input data.¯ ¯ ¯ ¯ ¯¯H /H (or H /H ) and s /S (Willmott, 1984). In locations where empirical data are unavail-b bc bn bnc

Accordingly, Willmott’s absolute-difference-based able — that might otherwise permit calculating aindex of agreement (d) (Willmott et al., 1985), and b — it is proposed that values of a50.0 and

¯ ¯the mean squared error (MSE), and the systematic b51.0 be used to estimate H and H frombc bnc

mean squared error (MSE ) were computed for observed irradiation and sunshine duration. Thiss

the irradiation and sunshine ratios at each site and approach would avoid any bias that might be†for the combined sites (Table 2). inherent in the coefficients in Table 2 or in

At individual sites, d ranges between 0.720 (at Louche et al. (1991).¯Montreal) and 0.833 (at Toronto); for all sites, In short, H can be estimated frombc

d 5 0.806. For Albany, Montreal, Toronto, and21¯ ¯ ¯ ¯H 5 H S (s ) (12)¯ ¯ bc bVancouver, the systematic error in H /H orb bc

¯ ¯H /H is close to, or less than, half of the total ¯bn bnc and H can be estimated frombncerror. For the combined sites, almost all (91.48%)

21¯ ¯ ¯ ¯H 5 H S (s ) . (13)of the error is unsystematic. This is also apparent bnc bn

in Fig. 6 where there is a fairly even distributionof points about the perfect-prediction line. At

5. CONCLUDING REMARKSUccle, in contrast, 96.81% of the total error issystematic; this is also illustrated in the relatively Eqs. (12) and (13) permit the estimation oflow d value for this site (0.758) and in Fig. 4 potential clear-sky beam irradiation from mea-

sured irradiation and sunshine duration. The po-tential flux density can, in turn, be used as input

†See Appendix A for a discussion of error measures. to turbidity models. Power and Willmott (2001),

Estimating clear-sky beam irradiation from sunshine duration 223

radiation and dust in the air. Geografiska Annaler 2, 156–¯for example, estimated H from Eq. (12). Usingbc 166.Power’s (2001) turbidity model, they then pro- ˚ ¨Angstrom A. (1956) On the computation of global radiationvided time series of total column aerosol over from records of sunshine. Arkiv. Geof. 2, 471–479.

Benson R. B., Paris M.V., Sherry J. E. and Justus C. G. (1984)South Africa. In addition to turbidity applications,Estimation of daily and monthly direct, diffuse and global

Eqs. (12) and (13) have utility in other solar solar radiation from sunshine duration measurements. Solarenergy applications where the monthly-averaged Energy 32, 523–535.

Gueymard C. (1993) Mathematically integrable parameteriza-potential irradiation is of interest.tion of clear-sky beam and global irradiances and its use indaily irradiation applications. Solar Energy 50(5), 385–397.

Gueymard C. (1999) Unpublished data provided by personalAcknowledgements—The author thanks Christian Gueymard, communication. Florida Solar Energy Center, 1679 Clear-Cort Willmott, and Brian Hanson for their assistance with this lake Rd, Cocoa, FL 32922-5703, USA. Present address: 174research, as well as three anonymous reviewers for their Bluebird Lane, Bailey, CO 80421, USA.helpful comments and suggestions. Financial support from the Gueymard C., Jindra P. and Estrada-Cajigal V. (1995) AGeography and Regional Science Program of the National ˚ ¨critical look at recent interpretations of the AngstromScience Foundation (SBR 97-09667) is gratefully acknowl- approach and its future in global solar radiation prediction.edged. Solar Energy 54(5), 357–363.

Hamdan M. A. and Al-Sayeh A. I. (1991) Diffuse and globalsolar radiation correlations for Jordan. Int. J. Solar Energy

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˚ ¨Hinrichsen K. (1994) The Angstrom formula with coefficientswhere a value of 1.0 indicates perfect agreement having a physical meaning. Solar Energy 52(6), 491–495.between two variables and 0.0 describes complete Iqbal M. (1979) Correlation of average diffuse and beam

radiation with hours of bright sunshine. Solar Energy 23(2),disagreement. The d index used here is equivalent169–173.to d in Willmott et al. (1985). The mean squared1 Iqbal M. (1983). An Introduction to Solar Radiation, Academ-

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Power H. C. (2001) Estimating atmospheric turbidity fromleast-squares regression between P and O . Thei iclimate data. Atmos. Environ. 35(1), 125–134.systematic error (MSE ) is the sum of the additives Power H. C. and Willmott C. J. (2001) Seasonal and inter-

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